Hi,
a) with
f[x_] := x^2 + NIntegrate[Exp[-t^2], {t, -Infinity, x}]
FindMinimum[f[x], {x, -2, 2}]
I got
{0.666069, {x -> -0.419364}}
with out an error message. Just use two values.
b) Is it so hard to compute D[Integrate[g[t],{t,-Infinity,x}],x] and
put it into the Gradient option of FindMinimum[] ?
(Abramowitz, Stegun, formula 3.3.7 ?)
D[#,c]& @ Integrate[f,{x,a,b}]==
f[b,c]*D[b,c]-f[a,c]*D[a,c]+Integrate[D[f,c],{x,a,b}]
Regards
Jens
Johannes Ludsteck wrote:
>
> Dear MathGroup Members,
> Unfortunately I got no answer when I sent the question below last
> week to the mathgroup mailing list. Since I think that the problem
> is not a very special one but a general problem of the way how
> Mathematica treats numerical integrals in the computation of
> gradients, I retry to get an answer.
> I want to minimize a complicated function which contains
> numerical integrals. Since the function is too complicated for a
> direct demonstration, I give a simple example which makes the
> structure of the problem clear:
>
> The (example) function to be minimized is:
> f[x_] := NIntegrate[g[t], {t, -Infinity, x}]
>
> (g is a known function; however symbolical integration is
> impossible).
>
> When I request numerical minimization of this function by typing
>
> FindMinimum[f[x],{x,1}]
>
> Mathematica gives me the following error message:
>
> FindMinimum::fmgl: Gradient {Indeterminate} is not a length 1
> list of real numbers at {x} = {1.}.
>
> Appearently, Mathematica is not able to find the gradient
> symbolically. A simple solution would be to define f using Integrate
> (without prefix N) and to wrap it with N[ ]:
>
> f[x_]:= N[ Integrate[g[t], {t,-Infinity, x}] ]
>
> However, since the function contains some hundred terms,
> evaluation of the function takes several minutes. (Mathematica then
> tries to find the integral symbolically before applying the numerical
> integration procedure.) This makes optimization impracticable.
> (the function I want to optimize has about 40 variables!).
>
> Are there any suggestions how to avoid computation of the gradient
> manually? (minimization algorithms which don't use the gradient
> are impracticable.)
> I.e. how can I tell Mathematica to use the first definition
>
> f[x_] := NIntegrate[g[t], {t, -Infinity, x}]
>
> for evaluation of the function and the second
>
> f[x_]:= N[ Integrate[g[t], {t,-Infinity, x}] ]
>
> for the computation of the gradient.
>
> Thank you
>
> P.S If you want to reproduce the error message, you can use a
> simple definition:
>
> f[x_]:= x^2 + NIntegrate[ Exp[-t^2], {t, -Infinity, x} ].
>
> Johannes Ludsteck
> Centre for European Economic Research (ZEW)
> Department of Labour Economics,
> Human Resources and Social Policy
> Phone (+49)(0)621/1235-157
> Fax (+49)(0)621/1235-225
>
> P.O.Box 103443
> D-68034 Mannheim
> GERMANY
>
> Email: ludsteck at zew.de